Question

Substitute the slope for m in our equation, or the rate of change of the height of the tree.
x
y
(4, 120)(7, 150)
0
1
2
3
4
5
6
7
8
9
10
0
15
30
45
60
75
90
105
120
135
150
165
180
Number of years
Height (inches)
y= mx + b
y=
x+b

Slope = 10

Answer 1

To substitute the slope \( m \) into the linear equation \( y = mx + b \), we first identify the values we have. From your provided points, we can find the slope:

The two points are:
- \( (4, 120) \)
- \( (7, 150) \)

The slope \( m \) is calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the values from the points:
\[
m = \frac{150 - 120}{7 - 4} = \frac{30}{3} = 10
\]

You mentioned that the slope is \( 10 \). We can now substitute \( m \) into the equation \( y = mx + b \).

So, we have:
\[
y = 10x + b
\]

Next, we need to determine the \( y \)-intercept \( b \). To do this, we can use one of the points provided (let’s use \( (4, 120) \)) to find \( b \):
\[
120 = 10(4) + b
\]
\[
120 = 40 + b
\]
\[
b = 120 - 40 = 80
\]

Now we can write the complete linear equation:
\[
y = 10x + 80
\]

Thus, the height of the tree in inches can be represented by the equation:
\[
y = 10x + 80
\]

In this equation:
- \( y \) is the height of the tree in inches,
- \( x \) is the number of years since the base year (the starting point),
- The slope \( 10 \) indicates that the tree grows \( 10 \) inches for each year that passes,
- The \( y \)-intercept \( 80 \) indicates that at year \( 0 \) (the base year), the height of the tree is \( 80 \) inches.